Dynamics of Gas Bubbles Encapsulated by a Viscoelastic Fluid Shell Under Acoustic Fields

J. Jimenez-Fernandez

Summary The dynamics of a gas-filled microbubble encapsulated by a viscoelastic fluid shell immersed in a Newtonian liquid and subject to an external pressure field is theoretically studied. The problem is formulated by considering a nonlinear Oldroyd type constitutive equation to model the rheological behavior of the fluid shell. Heat and mass transfer across the surface bubble have been neglected but radiation losses due to the compressibility of the surrounding liquid have been taken into account. Bubble collapse under sudden increase of the external pressure as well as nonlinear radial oscillations under ultrasound fields are investigated. The numerical results obtained show that the elasticity of the fluid coating intensifies oscillatory collapse and produces a strong increase of the amplitudes of radial oscillations which may become chaotic even for moderate driving pressure amplitudes. The role played by the elongational viscosity has also been analyzed and its influence on both, bubble collapse and radial oscillations, has been recognized. According to the theoretical predictions provided in the present work, a microbubble coated by a viscoelastic fluid shell is an oscillating system that, under acoustic driving, may experience volume oscillations of large amplitude, being, however, more stable than a free bubble. Thus, it could be expected that such a system may have a suitable behavior as an echogenic agent.

1. Introduction

The dynamics of free and encapsulated microbubbles is in­volved in numerous areas and, specially, in the biomedical field, where microbubbles are proven to be very valuable tools as contrast agents for medical ultrasound diagnosis. Indeed, when irradiated by ultrasound fields, these encap­sulated bubbles, known as ultrasound contrast agents, en­hance the blood-tissue contrast and thereby improve con­siderably the quality of ultrasonic images [1, 2]. More recently, they have also been employed for non-invasive therapy and targeted therapeutic drug delivery [3, 4]. Usu­ally, they consist of a gas core of low solubility which is stabilized against dissolution and coalescence by means of a thin layer of albumin, polymer or lipid material. The me­chanical properties of this coating play a dominant role on the overall dynamic behavior of the encapsulated bubble and, therefore, an adequate description of the rheological nature of the shell is a basic aspect of the theoretical anal­ysis.

A great amount of work has been devoted to study the dynamics and sound emission of gas filled encapsu­lated bubbles immersed in a liquid and irradiated by an acoustic field. Comprehensive reviews have been recently

published [5, 6]. In a first phenomenological approach, as those followed in pioneer works [7, 8], a generalized Rayleigh -Plesset equation was formulated by including in the analysis elastic and viscous properties of the encapsu­lating layer. These coating mechanical properties were in­troduced by adding to the usual surface tension coefficient, shell elasticity and shell friction terms in the normal stress condition at the gas liquid interface. Other studies have fol­lowed a more rigorous analysis based on basic principles of continuous mechanics. A viscoelastic solid layer of fi­nite thickness was considered by Church [9], where the shell was modeled by means of the Kelvin-Voigt rheolog­ical equation. The Church model was subsequently sim­plified in the limit of zero thickness layer by Hoff et al. [10] in order to describe the behavior of bubbles encap­sulated by polymeric shells. The dynamics of an encapsu­lated gas bubble surrounded by a compressible viscoelas­tic fluid was investigated by Khismatullin and Nadim [11]. In this work, the shell was also modeled by the Kelvin-Voigt rheological equation and it was shown that the elas­tic and viscous properties of the shell are dominant over those of the host fluid.

Some approaches including non-linear constitutive equations to describe the rheological behavior of the shell have also been considered. The Mooney-Rivlin consti­tutive law along with the Skalak model were investi­gated by Tsiglifis and Pelekasis [12] in order to describe strain-softening as well as strain-hardening elastic behav-

iors respectively. The numerical results provided in that work, predict striking differences between both behaviors. The Mooney-Rivlin model has been also used to predict the behavior observed in acoustic experiments on lipid-shelled microbubbles [13].The constitutive equation of an isotropic hyperelastic neo-Hookean material has been considered by Allen and Rashid [14] to analyze the me­chanical problem. More recently, the acoustic backscatter of bubbles encapsulated by a solid coating modeled by this Theological equation has been investigated by Jime­nez-Fernandez [15] in order to evaluate harmonic reso­nance frequencies in nonlinear conditions. Alternative ap­proaches where the coating is described as a surfactant layer, have been proposed [16,17,18,19]. The assumption of an homogeneous and isotropic layer has been revised by Chatterjee and Sarkar [17], and Sarkar et al. [18]. They have proposed a different approach based on interfacial models with intrinsic surface rheology. In the same line, in the model of Stride [19], it is assumed that the bubble is encapsulated by a homogeneous molecular monolayer with surface tension and interfacial viscosity depending on the surface molecular concentration.

As pointed out by Doinikov and Bouakaz [5] the inter­facial models described above, as well as, those based on the Kelvin-Voigt equation, lead to similar results for oscil­lations of infinitesimal amplitude (linear domain). Thus, the constant shell parameters have been usually estimated by fitting the theoretical results provided by linear anal­ysis, with experimental data corresponding to frequency-dependent attenuation or radius time curves.

Nevertheless, these theoretical approaches have been challenged by several experimental observations, some of which, are even detected for acoustic pressures as low as 10 kPa. Among these, it should be firstly quoted the effect known as "compression-only" behavior, where the bub­ble experiences a noticeable compression phase but hardly expands [20]. Another effect observed is the "threshold­ing" behavior, that is, the existence of an acoustic pressure threshold for the onset of bubble oscillations [21, 22, 23]. Finally, other not much explained effect is the experimen­tally observed dependence of shell elasticity and shell vis­cosity on the initial bubble radius [16, 24, 25]. These ex­perimental results have motivated a revision of previous theoretical analysis and more complex rheological models have been proposed. Some of the experimental observa­tions quoted above have been appropriately described by the model proposed by Marmottant et al. [26], a heuris­tic model based on the behavior of phospholipid mono­layer coating with surface tension depending on bubble area which takes into account shell buckling and rupture. By assuming that a lipid coating may be a material which exhibit both, shear thinning and strain-softening behavior, Doinikov et al. [25] have developed a nonlinear model which predict, in good agreement with experimental re­sults, compression only behavior. Two nonlinear interfa­cial models have been formulated by Paul et al. [27] which under some conditions also predict compression only be­havior. Sijl et al. [28] have shown with the aid of the

Marmottant model through a weakly nonlinear analysis that compression only behavior may result from a sudden change of shell elasticity with bubble radius. The Marmot­tant model has been recently extended by considering the nonlinear Cross-law for the shell viscous term [29] in or­der to introduce shear-thinning behavior.

For liquid coating the problem has been less explored. For a class of encapsulated microbubbles used in therapeu­tic applications, a Newtonian fluid layer was considered by Allen et al. [30]. For lipid coating, the equation of a lin­ear Maxwell fluid has been considered by Doinikov and Dayton [31] in an attempt to explain some experimental results which show a considerable increase of the reso­nance frequency of encapsulated bubbles compared with free bubbles.

In this work, the dynamics of gas-filled bubbles encap­sulated by a viscoelastic fluid shell immersed in an infinite Newtonian liquid and subject to an external pressure field is theoretically investigated. The problem is formulated by considering for the fuid shell, a nonlinear Oldroyd type constitutive equation with an interpolated time derivative [32, 33]. Heat and mass transfer across the encapsulating layer has been neglected but radiation losses due to the compressibility of the surrounding liquid have been taken into account. Bubble collapse under sudden increase of the external ambient pressure as well as nonlinear radial oscil­lations under monochromatic acoustic fields are analyzed. The objective of the work is to determine the quantitative influence of the elastic as well as viscous properties of the fluid shell on bubble collapse and bubble radial oscilla­tions.

The numerical results obtained show that fluid elastic­ity enhances considerably the amplitude of the radial os­cillations. In fact, chaotic behavior is predicted for mod­erate pressure amplitudes although critical values for bi­furcation and chaos are larger than those corresponding to a free bubble. On the other hand, the role played by the elongational viscosity in both bubble collapse and bubble oscillations has been investigated.

Besides eventual applications in other areas, the re­sults obtained in the present analysis may be of partic­ular interest in biomedical applications. Indeed, accord­ing to the theoretical predictions provided in this work, it may be concluded that a bubble encapsulated by a vis­coelastic fluid shell could have an excellent behavior as an echogenic agent.

The full paper is organized as follows: In section 2 the equations governing the radial motion are formulated ac­cording to the Oldroyd rheological model. Results are pre­sented in section 3. Section 3.1 is devoted to investigate the bubble collapse under a sudden increase of the ambi­ent pressure. In section 3.2 bubble oscillations driven by a monochromatic acoustic wave are analyzed. Dynamic be­havior and transition to chaos are described in section 3.3. In section 3.4 the dependence of the maximal amplitude expansion on the governing parameters is determined. Fi­nally in section 4 the main conclusion of the work are sum­marized.

2. The mathematical model

2.1. The equations for radial motions

Let us consider a gas bubble encapsulated by a viscoelastic shell and immersed in a Newtonian liquid of density pi and viscosity r\i. In a spherical coordinate system [r, 9, <p} with its origin at the center of the spherical bubble, the instantaneous inner and outer radii are denoted by r\ and r2 respectively. It is assumed that the volumetric bubble motion is purely radial so that spherical symmetry is held at any time. It is also considered that the encapsulating layer is incompressible and therefore,

r\-r\=I^-R\, (1)

where R\ and R2 are the initial inner and outer radii re­spectively.

Following a similar analysis to the one performed in [15], the equations of motion in the viscoelastic shell are integrated between r\ and r2 to obtain

Ps [<*></i) - <p(r2)] = Trr(r2) - Trr{ry) + T(ru t), (2)

where ps is the density of the shell, Trr is the rr-component of the stress tensor in the shell T, and cp(r) is the potential,

, , 1 d 2 {hr\f r dt 2n

(3)

where the dot denotes differentiation with time and T{r\) is the integral

r(n.o = 2 Trr — T99

dr. (4)

rrr, Tee are, respectively, the rr-component and the 99-component of the extra stress tensor T = T + p\ in the shell, p is the isotropic pressure and I is the identity tensor.

Trr{r\) and Trr(r2) are determined from normal stress balances at the inner and outer interface respectively. Thus, at r = r\,

Trr{rx) = -pg( / i ) + 2o\

(5)

where pg = pgo(r\/R\fr is the gas pressure inside the bubble, pgQ the gas pressure at the initial state, y the poly-tropic index and a\ the surface tension coefficient at the gas-shell surface. Likewise, a stress balance at the external interphase r = r2 gives

T„(r2) = TUri) • 2a2 (6)

where a2 is the surface tension coefficient at the shell-liquid surface and Tl

rr, the rr-component of the stress ten­sor in the liquid phase T'.r/r(r2) is determined by integra­tion of the motion equations in the external liquid phase. In the present analysis, heat transfer across the surface layer has been neglected but compressibility effects in the liquid

phase have been taken into account. Indeed, linear analy­sis for encapsulated microbubbles have shown that thermal and acoustic damping are negligible with respect to vis­cous damping [9, 11]. Nevertheless, as it has been noted by Doinikov-Boukaz [5], no available analysis provides whether these conclusions on oscillations of infinitesi­mal amplitude hold for non-linear oscillations. Moreover, apart from the relative importance of thermal and acoustic damping on the total damping constant, it should be em­phasized, that for micrometer bubble radius size and mega­hertz frequencies, the bubble shell wall velocity can reach values high enough to invalid the assumption of an incom­pressible liquid host. In other words, the effects of acoustic radiation may be of substantial importance for large ampli­tude bubble oscillations. Consequently, in the present work a boundary layer analysis analogous to those performed by Prosperetti and Lezzi [34] for a free bubble and by Khis-matullin and Nadim [11] for an encapsulated bubble, has been carried out in order to include compressibility effects in the liquid phase. Accordingly, the following expression for Tl

rr(r2) has been obtained,

TlM) = ~4t1i-~ Pi<P(r2) - Pao + -—Mr2), (7) ri cs df2

where p^, is the pressure far away from the bubble (a con­stant value or a monochromatic wave) and cs the sound velocity in the liquid phase. Finally, by substitution of ex­pressions (3) to (7) into expression (2) the following gov­erning equation for the bubble radial motion is obtained,

Psr\r\

+Psfl

Pi

pscs -ri/2

Pi 1 d

l + / i

3 r\ - + ^ ( 2 -) - ^-—rixi 2 2r\ Pscs 2 dt

(P-P^) + T--T- (P-Poo), pscs df 1 + x\

where

JiCn) • psr\

ri Xiir\)

1 + X\r\/f\

(8)

(9)

and

(10) (2oi 2a2\ r2

p=pg- + - 4t]t— + T. \ n r2 ) r2

2.2. The constitutive equation for the shell

The function Y(r\, t), as defined in (4), should be deter­mined from a constitutive equation for the extra stress ten­sor T. In this analysis, a nonlinear Oldroyd type equation [33] will be considered. This rheological model is defined by the equation

DT T + k\ ( — a(V • e + e • r)

Dt

= 2t] e+ 2t]A2 I - ^ - 2ae • e ) , Dt

(11)

where r\ is the shell viscosity, e = jCVv + (Vv) r ) the rate of strain tensor and Vv the velocity gradient tensor. k\ is the stress relaxation time and I2 the retardation time. D/bt is the Jaumann derivative given for an arbitrary sec­ond order tensor 0 by: Dfi/Dt = dfi/dt + vV/? + fiw - wfi where w is the vorticity tensor that for this irrotational flow vanishes. The term in large brackets in expression (11), known as the Gordon-Schowalter derivative, is an invariant derivative for any value of a in the interval: - 1 < a < 1. [35]. Depending on the values assigned to the parameter a, different rheological responses are pre­dicted. For a = 1 expression (11) provides the Oldroyd-B model which reduces to the upper convected Maxwell model (UCM model) if h = 0. The Oldroyd-B model ( a = 1 ), as well as the Oldroyd-A model (a = - 1 ) pre­dict a constant shear viscosity and an unbounded elonga-tional viscosity, however, for intermediate values of a the above model exhibits both extensional viscosity depend­ing on the extensional rate and shear thinning viscosity.

2.3. The equations of motion for the Oldroyd model

It is convenient decompose the extra stress tensor in the shell Tin the form T = TS + T ^ , where TN = 2riNe, is a Newtonian contribution with ijN = r\hlh and TS satisfies the equation

Ii>r \ rs + l i — a(ts • e + e • rs) ) =2rjse, (12)

\ Dt J where r/s = r\ - r\N = r\{\ - A2/Ai). For polymeric so­lutions, f]N may be regarded like the viscosity of a New­tonian solvent and r\s the polymer viscosity, so that»/. = fls + IN is the total solution viscosity. According to the above decomposition, equation (4) may be written in the form

r ( r i , 0 = - 4 i ^ ^ - + S ( r i , 0 . (13)

J?2 _ -^i is the constant volume of the shell

r2n

where V and S(ri, t) is defined as

r <r - T. S{n,t) ee dr. (14)

Consider the Lagrangian coordinate transformation,

3h = r3 - r (15)

After this transformation, expression (14) is written

S{n,t) vee Ar

•V/3 Ts _ , vee

3h + r? Ah, (16)

and Trr and Tee are obtained as solution of the equations

' d <

~At + l i • ae(tK 2rjse{t), (17)

Tee + ^1 At + ae(t)r* = -t]se(t), (18)

where

e(0 2r\r\

3h + r\ (19)

Integration of the above equat ions from f = 0 to f = t,

leads to

and

4?7p

' M

2fb A

t-r2M')h{t') W d f (2°) (3h + r])~

t-r2M')h{t') (3h + r\)2-

(3A + r J ) f + 1

Replacing above expressions in equation (16) gives

df'. (21)

S{n,t) M

„2/J e T ^ ( O r i ( 0 [ * i - Y 2 ] d ( ' . (22)

where

Yi = r/3

0 3h + r\ •Ah (23)

4a[r3(t)-r3(t')] 4(0

. ' 2 ( 0 . . Aw.

and

¥2 = v/3 vee 0 3h + r\

•Ah (24)

2a\r\{t) - r\(t')\

2a '' r\(ty

A«'\

2a

For any value of the parameter a, expressions (13), (22)-(24) introduced in equations (8)—(10) lead to an integrod-ifferential equation for r\ whose resolution presents diffi­cult problems for arbitrary values of the governing param­eters. However, as it has been previously shown by Shul-man and Levitskii [36] and Jimenez-Fernandez and Cre-spo [37], for a = 1/2 or a = 1 (Oldroyd-B model) this integrodifferential equation may be reduced to an ordinary differential system. Indeed, for a = 1/2, equation (22) may be written in the form: S(t) = Si(t) + ^ ( f ) , where

Si(t) = -4t1P

Mrt(t) J e-1! ri(t')h(t')At' (25)

and

Si{t) = 4lP

kriit)

" ^r\{t')h{t') Qh -———df'.

0 ri(t') (26)

Derivation with respect to time and ulterior application of the Leibniz theorem reduce the above expressions to the ordinary differential system

5i = - ( f + 2 - W - 4 r - -W i n I h n

S2 = - ( ^ + 2 - ) ^ 2 + 4 - ^ - . W i r2 I h r2

(27)

(28)

Likewise for a = 1 (Oldroyd-B) it is obtained S{t) Sn(t) + 5*12(0 + S2\(t) + S22(t), where

W i n I k\ n

Sn = -\

>21

>22

| + 2 £ L ) 5 l 2 - 2 ^ . X\ r\ J X\ r\

M r2 J M r2

f+2^V-+2f-k r2 I h r2

(29)

(30)

(31)

(32)

where/i(xi) = 1 + / (x i ) - XMF(x\)x\,

/2(xi) = \ + /(xi)(2 - i L ) - l-KM-^xxF(xx), (38) Zx2

,xi 1 + ( ^ - 1 } ( t ) 4

/(*) = (,-!)-, ^ = _ _ _ A _ / . .(39)

/ ( x i ) = ?( — 3y

2 Wei + We2\

Xi X2

4 x2 4e ±i(d3 - 1) J?e/ x2 Res x 3 X i

+ 5**, (40)

and S* is given for the following systems depending on the values of the parameter a, for a =1/2:

s* = s* + s*2

^1 + 1 ^ + 2 - K 1 \ De xi / 1

1 x2 Sn + l — + 2 — )s: De x2

,,(1 - e ) * i DeRes x\

,(1 - e ) * 2 DeRes x2'

(41)

(42)

(43)

The above reduction to a differential system may be also accomplished for a = - 1 (Oldroyd-A). Indeed, following a similar procedure it has been obtained

for a = 1 :

O - O n + o 1 2 + o 2 1 + o 2 2 ,

>12

>21

5*22 =

l i r2 / l i r2

\ l i r2 / h r2

(33)

(34)

(35)

(36)

The system (8)-(10) with T(ru t) defined in (13) provides the governing equations for the radial bubble motion. The function S(r\,t) is determined for a = 1/2 by the dif­ferential system (27)-(28), for a = 1 by the differential system (29)-(32) and for a = - 1 by the differential sys­tem (33)-(36), respectively. These systems are written in dimensionless form by introducing the scales: length: R\, time: R\ yjps/pc, and pressure pc : i) a constant value far away from the bubble p^ for the collapse problem or ii) the ambient pressure po when the external pressure is a monochromatic wave. Accordingly, the following non di­mensional equations are obtained,

/l(Xi)XiXi +f2(Xi)Xl =

d xi(p*-p*m) (37)

P*-P*M+KM

(1 - e ) x i

DeRes x2'

and for a = - 1 :

11 \ De xl I u DeResxi

S,n + ( - — + 2 — )SU = -2

S2l +

S22 +

Xi

x2

De x2 S* = 2

De x2 22

,(1 -e)xi DeRes x\

(1 -e)x2

DeRes x2'

,(1 -e)x2

DeRes x2

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

dt* l + / ( x i )

Stars, which will be suppressed hereafter, denote dimen­sionless quantities, xi = r\/R\, x2 = r2/R\, K = j - ,

d = R2/R1, M = ^/p~c/cs^fp~s is the Mach number, <1 = Pgo/Pc, Wei = 2(ji/Ripc, We2 = 2a2/R\pc are the Weber numbers for the internal and external interfaces re­spectively, Rei = piR\*Jp~c/(riiiJfc) is the Reynolds num­ber in the liquid phase, Res = Ri sfpTihl*] the Reynolds number in the shell, Be = k\ /tc the Deborah number and e = k2/ k\.

3. Results and discussion

3.1. Bubble collapse

Consider that the bubble, at rest at the external pressure Po for instant times prior to zero, is driven out of equi­librium when the external pressure is suddenly increased to a constant value px, so that the bubble evolves to a new equilibrium state. In order to analyze this dynamic process, the system (37)-(40) along with the appropriated system associated to S(xi, t) for any value of a, will be numeri­cally solved along with the initial conditions: xi(0) = 1, x(0) = 0, S(xi,Q) = 0. In the following, it will be con­sidered that the host fluid is water and, therefore, results will be obtained by fixing the values: pi = 1000Kg/m3, r\i = 0.001 Pas, cs = 1450m/s and the ambient pres­sure po = 1.013 x 105 Pa. In accordance with the ex­perimental data available for ultrasound contrast agents provided in previous works on encapsulated bubble dy­namics [9, 11], it will also be considered as fixed val­ues in ulterior calculations, the equilibrium inner radius Ri = 2fim, the polytropic index y = 1.4, the shell den­sity /?j = 1100 Kg/m3, and the surface tension coefficients: o\ = 0.04 N/m, a2 = 0.005 N/m. Regarding the shell vis­cosity r\ and the relaxation time k\, the estimated values provided by Doinikov and Dayton [31] for lipid-shelled microbubbles will be here adopted. As it is described in this work, the relaxation time k\ may be estimated by the quotient between the shell viscosity r\ and the shell elastic­ity modulus G corresponding to the Kelvin-Voigt model. In turn, the parameters G and r\ may be related, in the lin­ear domain, with the experimental data available for the dilatational viscosity and the elastic compression modu­lus involved in interfacial models. Accordingly, shell vis­cosity values in the range: 0.5 - 1.75Pas, and relaxation time values in the range: 0 . 0 1 - 1 |is will be, respectively, considered. The retardation time k2 appears in the analy­sis through the quotient k2/k\. This one must be greater than 1/9 to assure that the shear stress increases mono-tonically with increasing shear rate for steady shear flows [32]. Thus, a minimum value e = k2/k\ = 1/9 will be considered for all calculations.

In this section (Figures 1-5) results will be obtained for a pressure ratio q = po/Pca = 0-01. a shell viscosity r\ = 0.5 Pa s and a layer thickness R2 - R\ = 2 nm, except for Figure 3b, where r\= 1.75 Pa s and R2 - R\ = 10 nm.

In Figure 1 results are shown for the collapse of a free-bubble, a Newtonian shell (De = 0) and a viscoelastic fluid shell with De = 1 and a = 1/2. To illustrate the contrast with an elastic solid layer, results for a Kelvin-

1.0

0.8

n/R, 0.6

0.4

0.2

0

\ * . \ s''~~ " - " - - —

\ Y. '• / -4— -~~~^_

\

1 2 3 4

Dimensionless time

Figure 1. Bubble collapse: normalized inner radius ri/Ri vs di­mensionless time. Solid line: viscoelastic fluid shell with De = 1, dashed line: Newtonian fluid shell (De = 0), dotted line: free-bubble, dash-dotted line: elastic Kelvin-Voigt shell with G = 50MPa.

Figure 2. Bubble collapse: trayectories in the phase space. Solid line: viscoelastic fluid shell with De = 1, dashed line: Newtonian fluid shell (De = 0), dotted line: elastic Kelvin-Voigt shell with G = 50MPa.

Voigt shell according to the formulation outlined in a pre­vious work [15], with a shear modulus G = 50MPa have also been included in the same figure. As it may be ob­served at first sight, for the parameter values considered, an oscillatory behavior similar to that observed for free bubbles, is predicted for a viscoelastic shell whereas the collapse is monotonous for a Newtonian shell. Note, how­ever, the damping influence of the shell which produces a clear reduction of the amplitude of radial oscillations. This time evolution to new equilibrium states is shown in Figure 2 where trayectories in the phase space have been plotted. In the elastic case the bubble follows a damped oscillatory trayectory towards a equilibrium state close to the initial one. By contrast, for a viscoelastic fluid shell the integrated normal stress difference S tends to zero for

t » 1 and, therefore, the normalized inner radius tend to the Newtonian value for large times given as the solution of the algebraic equation

3y 2Wel

x1

2We2

(xl+d3-iy/3 - 1 = 0 . (52)

The more relevant parameter to characterize the elastic­ity of the fluid layer is the Deborah number, defined by the quotient between the relaxation time of the fluid and the characteristic time of the process. His explicit influ­ence is illustrated in Figure 3ab, where normalized radius versus normalized time is plotted for De = 0.1,1,5, a shell Reynolds number Res = 0.04 (ij = 0.5Pas) and a layer thickness Ri - R\ = 2nm (Figure 3a) and for De = 0.1,0.5,1, a shell Reynolds numbers Res = 0.008 (fj = 1.75Pas) and a layer thickness Ri - R\ = 10nm in Figure 3b. As the Deborah number increases, the vis-coelastic character of the fluid is enhanced and, corre­spondingly, oscillatory collapse is more clearly observed, as it is shown in Figure 3a. It may be also clearly recog­nized in Figure 3b, that at the early stages of the collapse process, the collapse rate is increased as De increases. This is just the opposite behavior to one would expect, tack­ing into account that the fluid flow induced by the bubble collapse is an unsteady uniaxial extension flow for which, the extensional resistance should be larger in a viscoelas-tic fluid layer than that corresponding to a Newtonian fluid layer. In fact, it is easy to show that, in steady uniaxial flow, the constitutive equation (11) predicts an increase in the elongational viscosity as De increases. This situation reproduces the one found for bubble collapse in viscoelas-tic fluids [36, 37, 38, 39, 40, 41], where it is predicted that fluid elasticity accelerates the early stages of the collapse and retards the final stages of the process. Indeed, as de­scribed by Zana and Leal [39], the bubble is set into mo­tion from an equilibrium rest state and, initially, there is a period of stress growth where the instantaneous resistance to motion is lesser than the one corresponding to a steady uniaxial flow at the same elongational rate. Consequently, at the early stages of the process, the unsteady collapse rate overshoots the corresponding steady state value. By contrast, in the last periods of the collapse, a steady flow is approached and the influence of larger values of the elon­gational viscosity in viscoelastic fluid layers becomes evi­dent. It is wort mentioning, the similarity between bubble collapse of a free bubble in a viscoelastic medium and the collapse of a bubble coated by a viscoelastic layer in a Newtonian fluid.

Discrepancies with respect to the Newtonian behavior may be also quantified by means of the rheological pa­rameter e = hih, as it is decreased from e = 1 (Newto­nian fluid). Results for fixed De = 0.5, r\ = 0.5Pas, and the values: e = 1/9,0.3,0.6,1 are shown in Figure 4. Here again, oscillatory collapse is induced as e decreases. It may be also observed that at the early stages of the process, the collapse rate increases as e decreases, in contrast to the be­havior expected in view of the increase of the elongational

rj/JJ;

1.0

0.9

0.8

0.7

0.6

0.5

0.4

• - ^

\ (a)

\ \ \

\ - ' ' - ' ' " \

\\/ / v/

0.0

1.0

0.9

n'Ri

0.8

0.7

0.6

0.5

t.

V\ i

1.0 1.5 2.0 2.5 3.0 Dimensionless time

(b)

—-sN

^ ^ ^ - - - ^

3 4 Dimensionless time

Figure 3. Bubble collapse: influence of the Deborah number, (a) Res = 0.04, R2 - Ri = 2nm. Solid line: De = 5, dashed line: De = 1, dotted line: De = 0.1. (b) Res = 0.008, R2 - Rt = 10 nm. Solid line: De = 1, dashed line: De = 0.5, dotted line: De = 0.l.

0.5 1.0 1.5 2.0 2.5 Dimensionless time

Figure 4. Bubble collapse: influence of the parameter e = M/M • Solid line: e = 1/9, dashed line: e = 0.3, dotted line: e = 0.6, dashed line: e = 1 (Newtonian fluid).

viscosity as the parameter e is reduced. As noted previ­ously, this apparent anomalous behavior is a consequence of the overshoot of the unsteady collapse rate on the cor­responding steady value.

To finish this section, the role played by the rheological parameter a will be analyzed. The constitutive equation (11) predicts an enhancement of the elongational viscos­ity as the parameter a increases. Accordingly, the collapse radius should be reduced and the collapse rate increased as a growths. This is just the behavior that has been found

1.0 ; ^ ^

0.9 • \

0.8 : \

rxIRx : \ 0.7 ; \

0.6 • \

o.3 '• y

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Dimensionless time

Figure 5. Bubble collapse: influence of the parameter a. Solid line a = — 1 (Oldroyd-A), dashed line a = 1/2, dotted line a = 1 (Oldroyd-B).

Figure 6. Normalized inner bubble radius versus time in acoustic cycles, (a) solid line De = 1, dashed line De = 0.5, dotted line: Newtonian, (b) dotted line a = — 1 (Oldroyd-A ), dashed line a = 1/2, solid line a = 1 (Oldroyd-B).

as it is illustrated in Figure 5, where collapse curves are shown for a = 1 (Oldroyd B), a = 1/2, a = - 1 (Oldroyd A) and De = 1, r\ = 0.5 Pa s, respectively. This double de­pendence of the bubble dynamics on both, the parameter a and the Deborah number will be revisited and discussed in the next section.

3.2. Dynamic behavior under periodic forcing

In this section it will be considered that the bubble is subject to the action of a monochromatic wave px(t) =

Po + PA sin(<B£)> where pA is the acoustic pressure ampli­tude, a) = 2nf the angular frequency and / the acous­tic frequency. Numerical results will be obtained for mi-crobubbles immersed in water (data for the host fluid as in section 3.1) with an initial inner radius R\ = 2|jm, a layer thickness Ri- R\ = 2 nm (except Figure 6) and driven by an acoustic wave with / = 2 MHz, a typical value in di­agnostic applications. The rest of parameter values will be specified for each case. Bubble radius versus time curves are shown in Figure 6 for bubbles with a layer thickness Ri - R\ = 20nm, shell viscosity r\ = 1.5Pas, and pres­sure amplitude pA = 150 kPa. Curves for Deborah num­bers De = 0, 0.5,1, are shown in Figure 6a, respectively. It may be observed that, as it was described in section 3.1 for the collapse problem, fluid elasticity of the shell pro­duces a dramatic increase of the radial amplitude, specially for De = 1. The influence of the Theological parameter a is shown in Figure 6b where inner radius versus time is plot­ted for De = 1 and a = - 1 (Oldroyd-A), a = 1/2, a = 1 (Oldroyd-B). Note that as a goes from a = - 1 to a = 1, the amplitude of the maximal radial expansion is consid­erably increased. Note also that, conversely, the minimum radius for a = - 1 is smaller (larger in absolute value) than the corresponding to a = 1 with an intermediate value for a = 1/2. This finding that has been previously dis­cussed for the collapse problem, deserves special atten­tion. According to the predictions of the constitutive equa­tion (11) for steady extensional flows, the elongational vis­cosity increases with the parameter a in uniaxial flows and decreases with a in biaxial flow. Consequently, it should be expected that the elongational viscosity increases as the parameter a goes from a = - 1 to a = 1 in the uniaxial compression stages, and decreases in the biaxial expansion phases. Thus, minimum radii should be larger for a = 1 than those for a = 1/2, and a = -1,whereas the inverse behavior should be observed for the maximum radial am­plitudes. This is just the behavior predicted in the present analysis as it is illustrated in Figure 6b.

3.3. Regular and chaotic behavior

The dynamic behavior of an encapsulated microbubble with a shell viscosity r\ = 1 Pas and a = 1/2, has been analyzed for an acoustic forcing pA = 350 kPa, by increas­ing the Deborah number from De = 0 (Newtonian case) to De = 0.5. In Figure 7 normalized inner radius ri/Ri versus time (left column) as well as Poincare section plots (right column) are shown. In these ones, the normalized bubble wall velocity r\ptc/R\ is plotted versus the normal­ized radius r\p/R\, where r\p is the inner bubble radius at the end of each acoustic cycle. The results show doubling period and transition to chaos as the Deborah number is in­creased, a behavior that is more clearly illustrated in the bi­furcation diagram with the Deborah number as control pa­rameter shown in Figure 8 (data as in Figure 7). A similar behavior was predicted and described for a free bubble im-merged in a viscoelastic fluid [42, 43]. Recently, dynamic system theory has been applied in a comprehensive anal­ysis of the nonlinear response of encapsulated microbub-

J ' lCt) 2^0

496 497 1.0 1.5 2.0 2.5 *i(t)

/ i ( t ) ,

yi(t)

1.0 1.5 2.0 2.5 *i(t)

Figure 7. Left column: Normalized inner bubble radius ri/Ri versus time in acoustic cycles q = 1 Pa s, / = 2 MHz. pA = 350 kPa. Right column: Poincare section plot. From top to bot­tom: De = 0 (Newtonian case). De = 0.05 and De = 0.5.

2.0

x\p

1.5

1.0

•.

0.05

, lillliBlllgJili i « i !II| i -!-:::!::;

*"&-• : ,-• ••:• • . V - -..'• : • • ' . ' • . " ' - • :

•SillJiA . t | i t / .••• . . - , . ' iV f j I i

"' JS'illS^.iiliii.ii.l^^S:!1

0.10 0.15 0.20

Deborah number

Figure 8. Bifurcation diagrams with the Deborah number as con­trol parameter.

(a) 2.0 2.2 2.4 2.6 2.8

Pressure amplitude

Pressure amplitude

(c) Pressure amplitude

Figure 9. Bifurcation diagrams with acoustic pressure as control parameter, (a) viscoelastic fluid shell with De = 1 (b) Newtonian fluid shell, (c) free-bubble.

bles [44]. It is not the purpose of this work to perform an exhaustive study of the dynamic problem here considered and outlined in section 2, nevertheless, the results provided in this section, may illustrate the important role played by the fluid elasticity (Deborah number) in the nonlinear re­sponses of this complex system. Bifurcation diagrams with the pressure amplitude as control parameter are shown in Figure 9a for De = 1 and for De = 0 (Newtonian case) in Figure 9b. In order to accomplish a direct comparison, the bifurcation diagram of a free-bubble of the same size and driven at the same frequency is plotted in Figure 9c (data as in Figure 7 with r\ = 0.75Pas). Note that for a viscoelastic coated bubble the transition to chaos occurs at pressure amplitudes above those that produce chaotic os­cillations for a free-bubble. In other words, a microbubble encapsulated by a viscoelastic fluid shell experiences regu­

lar periodic radial oscillations at pressures that drive a free bubble into chaotic oscillations.

3.4. Maximum relative expansion

It has been shown in previous sections that under acous­tic driving, bubbles encapsulated by a viscoelastic fluid shell undergo radial oscillations of considerably large am­plitude, a result that may be of interest in diagnosis imag­ing. Accordingly, the relative importance of the governing parameters on the maximum amplitude expansion will be analyzed in detail. Thus, in this section, the quantitative influence of the driving pressure amplitude, the Deborah number, the shell Reynolds number and the shell thick­ness, will be subsequently estimated. In the following, numerical calculations will be performed by considering a = 1/2, and the fixed parameter values quoted in sec­tion 3.2.

1 0 A, ^ 1 5

pA(bar)

Figure 10. Normalized maximum radial amplitude r\ max/Ri ver­sus acoustic pressure PA- blue line: Newtonian (De = 0), green line: De = 0.1, red line: De = 1. Blue dashed line: Kelvin-Voigt solid (G = 166.7 MPa), red dashed line: free bubble.

1.8

1.7

„ 1.6

1 1-5

1.4

1.3

1.2

0.01 0.05

/ / / / / /

0.10 0.50 Deborah number

1.00 5.00

Figure 11. Normalized maximum radial amplitude r\ max/R\ ver­sus Deborah number: Dotted line: PA = 50kPa, dashed line: pA = 100kPa, solid line: pA = 200 kPa.

The maximal radial amplitude r\ max/ R\ as a function of the pressure pA for different values of De and a shell Reynolds number Res = 0.017 (»/ = 1.2Pas) is shown in Figure 10. For the sake of comparison, the corresponding curves for a solid shell (Kelvin-Voigt model) and a free bubble have also been plotted in this figure. The quanti­tative differences in radial expansion between solid coat­ing and fluid coating are evident. For liquid shells, the in­fluence of De becomes particularly apparent as the pres­sure amplitude increases. Note, however, the minute differ­ences that may be distinguished for De = 1, and De = 10 or larger values (not plotted) with respect to a free bubble. In fact, for any pressure amplitude, r\ max /R\ increases with De, reaches a maximum value for De « 1 which is close to the one corresponding to a free bubble and then decreases to an asymptotic value for De » 1, as it is illustrated in Figure 11 where r\max /R\ vs De is plot­ted for Res = 0.0 1(J/ = 2Pas) and pressure amplitudes pA = 50,100,200 kPa. The asymptotic behavior for large De may be determined from expressions (42)-(43) for the normal stress difference in dimensionless form. Indeed, for De » 1, the following analytical expression has been ob­tained:

S = 5*i + S2 = 2 ( l - £ ) / 1 DeRes \ v2 (53)

This expression shows that the asymptotic behavior of the normal stress difference for large De is controlled by (DeRes)~

l, so that, for moderate or large Res values, the viscoelastic coating approaches the behavior of a Newto­nian fluid shell (see expression 40).

The influence of the shell viscosity is illustrated in Fig­ure 12, where the maximal radial amplitude is plotted versus the Reynolds number of the layer Res for De = 0,0.1,1 and a pressure amplitude pA = 100 kPa. For small Deborah numbers, a significant dependence of the max­imal radial amplitude on Res is detected. However, for De = 1 or larger the influence of Res is relatively small. In other words, for Deborah numbers of the order of unity

1.55

1.50

1.45 85" %

a 1.40

1.35

1.30

1.25

/ / / / / /

/ / /

/ /' /

/ / 0.010 0.015 0.020 0.030 0.050 0.070 0.100

Rexnolds number

0.150 0.200 0.300

Figure 12. Normalized maximum radial amplitude rimax/Ri ver­sus the Reynolds number of the shell Res. Dotted line Newto­nian (De = 0), dashed line: De = 0.1, solid line: De = 1 and pA = lOOkPa.

Figure 13. Normalized maximum radial amplitude rimax/Ri ver­sus the thickness layer R2 — R\- Blue lines (lower solid and dashed curve) Newtonian (De = 0), red (upper solid and dashed curve) lines: De = 1. Solid lines correspond to pA = 200 kPa and dashed lines to pA = 100 kPa.

or larger, damping effects are irrelevant and the bubble dy­namics is governed by elastic effects.

Above results have been obtained by considering a small layer thickness R2 - R\ = 2nm, however, this pa­rameter may have, in the viscoelastic case, a consider­able quantitative influence as it is shown in Figure 13, where r\ max/R\ is plotted versus the layer thickness for Res = 0.01(J/ = 2Pa.s) pressure amplitudes pA = 100, 200 kPa and De = 0,1. Note that for small values of De or in the Newtonian case (De = 0), r\ max/R\ decreases smoothly as the layer thickness increases. However, for De = 1, and mainly at high pressure amplitudes, the max­imal radial expansion remains nearly constant for small thickness but decreases sharply above a given value of this one. A numerical example may illustrate this output: for De = 1 and pA = 200 kPa the maximum radial expansion i"imax/Ri = 1.8 for R2 - Ri below 25nm, and is abruptly reduced to r\max/R\ = 1.2 for R2 - Ri above 30 nm.

4. Conclusions

The dynamics of a gas bubble encapsulated by a viscoelas­tic shell and subject to the action of a external pressure field has been investigated. Bubble collapse under a sud­den change of the external pressure as well as radial vol­ume oscillations under an ultrasonic monochromatic field has been considered. The main results obtained may sum­marized as follows: • Oscillatory collapse is predicted for a viscoelastic shell

in conditions where the collapse of a Newtonian shell is monotonous. The elasticity of the fluid shell acceler­ates the early stages of the collapse and retards the final stages of the process. A behavior similar to the one ob­served for the bubble collapse of free-bubbles immersed in viscoelastic fluids.

• Shell elasticity enhances the amplitude of radial oscilla­tions which increases dramatically as the Deborah num­ber increases and become chaotic even for moderate values of the external pressure amplitude. Nevertheless, critical values for the onset of bifurcation and chaos for a viscoelastic shell, are smaller than those correspond­ing to a free bubble.

• The asymmetry of radial oscillations with respect to the initial equilibrium radius evidences the influence of the elongational viscosity. Thus, as the rheological param­eter a goes from a = - 1 (Oldroyd-A) to a = 1/2, and a = 1 (Oldroyd-B), maximal radii are clearly in­creased and minimal radii reduced. This result may be explained taking into account that, according to the pre­dictions of the Oldroyd model in extensional flows, as the parameter a increases, the elongational viscosity increases in the uniaxial flow of the collapsing phase whereas decreases in the biaxial flow corresponding to the expansion phase.

• For a value of the driving frequency / = 2 MHz, a typ­ical value in diagnosis applications, the maximal am­plitude of radial oscillations is observed for Deborah numbers around the unity. In the limit of very large Deborah numbers, an analytical expression for the nor­mal stress difference has been obtained. This expression

shows that the asymptotic behavior of the normal stress difference for large De is controlled by (DeRes)~

l, so that, for moderate or large Rs values, the viscoelastic coating approaches the behavior of a Newtonian fluid shell. For small Deborah numbers, elastic effects are re­duced as the Reynolds number of the layer decreases.

• The influence of the shell thickness has also been analyzed. It has been found that for small Debo­rah numbers, the maximum radial amplitude decreases smoothly for increasing coating thickness. However, for Deborah numbers of the order of unity, the maximal ra­dial expansion remains nearly constant for small layer thickness but decreases abruptly above a critical value of this one. This critical value increases as the ampli­tude of the driving pressure is increased.

According to the above theoretical predictions, it may be concluded that a microbubble coated by a viscoelastic fluid shell is an oscillating system that, under acoustic driving, may experience volume oscillations of large amplitude, being, however, more stable than a free bubble. Thus, it could be expected that such a system may have a suitable behavior as an echogenic agent.